I've decided to write up my answer given in the comments in details. The main tool I am going to use is decidability of the theory of real closed fields, and the fact $\mathbb R$ is a real closed field. What this means that if we are given a sentence in the first-order language of ordered fields (i.e. using $0,1$, addition, multiplication, inequality and quantifiers over numbers), we can algorithmically decide whether it's true in $\mathbb R$ or not.

That said, it's enough to give a procedure which, given some expression involving $+,-,\cdot,/,\sqrt{}$, will give us a formula which expresses whether it's zero. I will explain this through an example. Take $\sqrt{3+\sqrt{2}}-3$, the example from question.

- $\sqrt{2}$ is defined as $x_1$ satisfying $x_1\cdot x_1=1+1\land x_1>0$.
- $3+\sqrt{2}$ is defined as $x_2$ satisfying $\exists x_1:(x_2=1+1+1+x_1\land(x_1^2=1+1\land x_1>0))$
- $\sqrt{3+\sqrt{2}}$ is defined as $x_3$ satisfying $\exists x_2:\left(\exists x_1:(x_2=1+1+1+x_1\land(x_1^2=1+1\land x_1>0))\land x_3\cdot x_3=x_2\land x_3>0\right)$
- $\sqrt{3+\sqrt{2}}-3$ is defined as $x_4$ satisfying $\exists x_3:\left(\exists x_2:\left(\exists x_1:(x_2=1+1+1+x_1\land(x_1^2=1+1\land x_1>0))\land x_3\cdot x_3=x_2\land x_3>0\right)\land x_3=x_4+1+1+1\right)$
- Finally, $\sqrt{3+\sqrt{2}}-3=0$ is expressed as $\exists x_4:\left(\exists x_3:\left(\exists x_2:\left(\exists x_1:(x_2=1+1+1+x_1\land(x_1^2=1+1\land x_1>0))\\\land x_3\cdot x_3=x_2\land x_3>0\right)\land x_3=x_4+1+1+1\right)\land x_4=0\right)$

I hope the idea here is clear - we inductively give definitions for each subexpression, and then we state the element given by the final definition is equal to zero. It should be clear enough this translation can be done algorithmically, and hence equality of such expressions to zero can be algorithmically decided.

Let me make a quick note on practicality of this algorithm: it's nowhere near practical. Appealing to full decidability of theory of real closed fields is a major overkill. As you can read in the Wikipedia page linked above, the algorithm for deciding full theory is necessarily doubly exponential, so for the formula above, which is about 30 characters long, this would give us a bound on running time which is on the order of $2^{2^{30}}$, which is a number with 300 million digits!

However, the formulas we derive above are not some generic formulas - they are *existential*, i.e. don't involve any universal quantifiers. For those formulas there is an algorithm with exponential running time - faster, but still impractical, even for small formulas, see here for details.

In conclusion, this gives us that that the problem is solvable *in principle*. *In practice*, other algorithms are needed.

yes: more generally, there exist algorithms to perform exact computations on real algebraic numbers (i.e., compute their sums, differences, products, quotients, roots of polynomials, and perform comparisons for equality and order): see here and here for closely related answers, and references. $\endgroup$ – Gro-Tsen Feb 9 at 10:53